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I studied the classical Libor market model, where the dynamics of rate $F_k$ from time $T_{k-1}$ to $T_k$ are given by $$ dF_k(t)/F_k(t) = \sigma_k(t) dZ_k(t) $$ under the forward measures $Q^k$ (where we use $P_k(t)$, the bond that matures at $T_k$, as numeraire). Then, it follows by a change of measure approach that for $i=k-1$, the dynamics under this measure $Q^k$ are $$ dF_{k-1}(t)/F_{k-1}(t) = \sigma_{k-1}(t) \left(dZ_{k-1}(t) - \frac{\rho_{k,k-1} \sigma_k(t) F_k(t)}{1+\tau_k F_k(t)} dt \right). $$ The above formula are taken from the book of Brigo and Mercurio and there the general formulation for a rate $F_i$ with $i<k$, resp., $i>k$ can be found.

I was able to technically follow the derivation of the drift, but what is the intuitive understanding of it? Can we use a trading argument to understand this equation?

This might be quite basic but I did not find such an intuitive explanation in the literature. Any comment or reference is highly appreciated.

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Paying F(K-1) at a lag (K) is a delayed payment and involves a convexity adjustment that can be understood as a consequence of 2 parts - a "stochastic part" where if rates rise whenever F(K-1) is higher (and vice versa, if rates fall), we systemically lose out. The drift adjusts this valuation. This is why you have covariance between the forward and the bond Z(t,T(K-1),T(K)).

The denominator is the "deterministic" part of the convexity adjustment that just penalizes the delay based on the prevailing discount between k-1 and k.

You will now realize that the Radon-Nikodym derivative is ultimately a deterministic adjustment (A0/B0) times a stochastic adjustment (A(t)/B(t)), this is exactly that! So you can tie this back to the math.

Hope this helps.

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  • $\begingroup$ Thank you for this answer. Could you please elaborate a bit more on the delayed aspect? The SDE holds for $t< T_{k-2}$, so I don't see a delay here. If we discount with $P_t(k)$ then we discount with a bond with longer maturity, I understand that. I would appreciate more thoughts on that. Thank you! $\endgroup$
    – Richi Wa
    Commented May 13 at 8:19
  • $\begingroup$ F(k-1) sets at K-2 and is paid at K-1. That is its natural payment date. In measure Q-K, the relevant asset F(K-1) is valued as if paid at K. This is the payment delay adjustment - from K-1 to K. $\endgroup$
    – Arshdeep
    Commented May 13 at 10:08
  • $\begingroup$ Thanks, and how can I understand the numerator better? We systematically lose something proportional to $F_{k-1}*F_k$. If both rates are positive we always lose a bit (compared to the $P(k)$ bond, right?). Other question: is there any paper that discusses such intuition? Thanks! $\endgroup$
    – Richi Wa
    Commented May 13 at 10:51
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    $\begingroup$ quant.stackexchange.com/questions/43085/… $\endgroup$
    – Arshdeep
    Commented May 13 at 10:58
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    $\begingroup$ The third answer there provides intuition to the convexity adjustment, hopefully that should explain it. Numerator is just the covariance but because the rates are lognormal, it has the forward levels as well $dF=vol*F*dW$ $\endgroup$
    – Arshdeep
    Commented May 13 at 11:00

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